Birds only use a nest when they are having young. After that they don't use the nest any more, although some birds will reuse it for another clutch of eggs. Birds live mostly in tree branches and bushes. The tree that was cut down may have been one of the trees they were just used to being there. They will find another tree, don't worry, all will be well. Squirrels do live in their nests, and if it is destroyed, they will make another one elsewhere. They adapt very easily. Please read my profile.What do birds/squirrels do when their nest/ tree is cut down?
Generally, birds do not live in nests..they reproduce in them. Sure they may use the tree itself to sleep and roost in, but they should not be breeding so late in the year. If you live in the US, it is actually illegal to destroy a birds nest as they are protected by law. Other countries have similar laws, that is why it is suggest to only do trimming and cutting of trees during the fall and winter seasons as not to accidentally or purposely destroy nests. There are a FEW species that will build nests specifically to roost in, and magpies can be one of them, so if they were sitting on nests in the tree..they were empty nests, not containing eggs or chicks in which case you did not break the law, and in which case the birds will just move elsewhere and no real harm was done.
Find a new place to live.